The functional organization of neural processes is constrained by the brain's intrinsic structural connectivity, i.e., the connectome. Here, we explore how structural connectivity can improve the representation of brain activity signals and their dynamics. Using a multi-modal imaging dataset (electroencephalography, structural MRI, and diffusion MRI), we represent electrical brain activity at the cortical surface as a time-varying composition of harmonic modes of structural connectivity. These harmonic modes are known as connectome harmonics. Here we describe brain activity signal as a time-varying combination of connectome harmonics. We term this description as the connectome spectrum of the signal. We found that: first, the brain activity signal is represented more compactly by the connectome spectrum than by the traditional area-based representation; second, the connectome spectrum characterizes fast brain dynamics in terms of signal broadcasting profile, revealing different temporal regimes of integration and segregation that are consistent across participants. And last, the connectome spectrum characterizes fast brain dynamics with fewer degrees of freedom than area-based signal representations. Specifically, we show that a smaller number of dimensions capture the differences between low-level and high-level visual processing in the connectome spectrum. Also, we demonstrate that connectome harmonics capture more sensitively the topological properties of brain activity. In summary, this work provides statistical, functional, and topological evidence indicating that the description of brain activity in terms of structural connectivity fosters a more comprehensive understanding of large-scale dynamic neural functioning.